Kerry Back
BUSI 721, Fall 2022
JGSB, Rice University
Portfolio mean is weighted average of asset means
Portfolio std dev < weighted average of asset std devs
\(\sum_{i=1}^n w_{i}^2var(r_i)+2\sum_{i=1}^n\sum_{j=i+1}^n w_{i}w_{j}cov(r_{i},r_{j})\)
Let C = covariance matrix, w = vector of weights.
Portfolio variance is \(w^{⊤}Cw\).
Can calculate covariance matrix as SRS where S is diagonal matrix containing std devs and R is correlation matrix.
import numpy as np
# standard deviations
sds = np.array([0.20, 0.12, 0.15])
S = np.diag(sds)
# correlations
r12 = 0.4
r13 = 0.2
r23 = 0.3
R = np.identity(3)
R[0, 1] = R[1, 0] = r12
R[0, 2] = R[2, 0] = r13
R[1, 2] = R[2, 1] = r23
# covariance matrix
C = S @ R @ S
# portfolio
w = np.array(0.25, 0.25, 0.5)
# portfolio std dev
np.sqrt(w @ C @ w)Suppose asset B is risk free. Set \(r_{f}=r_{B}\).
Then, \(\small var(r_{p})=w_{A}^2var(r_{A})\) and \(\small stdev(r_{p})=w_{A}stdev(r_{A})\)
\[\small \bar{r}_p=w_{A}\bar{r}_{A}+(1-w_{A})r_{f}=r_{f}+w_{A}(\bar{r}_{A}-r_{f})\]
Portfolio risk premium is: \[\small \bar{r}_p-r_{f}=w_{A}(\bar{r}_{A}-r_{f})\]
So both risk and risk premium scale with allocation to risky asset.
The Sharpe ratio is the ratio of risk premium to risk:
\[\frac{\bar{r}-r_{f}}{\sigma}\]
When combining a risk-free asset with a risky asset, the Sharpe ratio is invariant to the risky asset allocation.
The Sharpe ratio of the portfolio is the Sharpe ratio of the risky asset.
Continuing with asset B = risk-free, we could take \(w_{A}>1\), meaning that \(w_{A}-1\) is borrowed.
Same formulas apply:
Borrowing rate will be higher than saving rate.
\(n\) risky assets. Allocation to risk-free asset is \(1-\sum_{i=1}^n w_{i}\).
Portfolio return is:
\[r_{p}=\sum_{i=1}^n w_{i}r_{i}+(1- \sum_{i=1}^n w_{i})r_{f}=r_{f}+\sum_{i=1}^n w_{i}(r_{i}-r_{f})\]
So, risk premium is weighted average of asset risk premia.